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Theorem chfacfscmulgsum 22813

Description: Breaking up a sum of values of the "characteristic factor function" scaled by a polynomial. (Contributed by AV, 9-Nov-2019.)

**Hypotheses**
Ref	Expression
chfacfisf.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
chfacfisf.b	⊢ 𝐵 = (Base‘𝐴)
chfacfisf.p	⊢ 𝑃 = (Poly₁‘𝑅)
chfacfisf.y	⊢ 𝑌 = (𝑁 Mat 𝑃)
chfacfisf.r	⊢ × = (._r‘𝑌)
chfacfisf.s	⊢ − = (-_g‘𝑌)
chfacfisf.0	⊢ 0 = (0_g‘𝑌)
chfacfisf.t	⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
chfacfisf.g	⊢ 𝐺 = (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))))
chfacfscmulcl.x	⊢ 𝑋 = (var₁‘𝑅)
chfacfscmulcl.m	⊢ · = ( ·_𝑠 ‘𝑌)
chfacfscmulcl.e	⊢ ↑ = (._g‘(mulGrp‘𝑃))
chfacfscmulgsum.p	⊢ + = (+_g‘𝑌)

**Assertion**
Ref	Expression
chfacfscmulgsum	⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ ℕ₀ ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))

Distinct variable groups: 𝐵,𝑛 𝑛,𝑀 𝑛,𝑁 𝑅,𝑛 𝑛,𝑌 𝑛,𝑏 𝑛,𝑠,𝐵 0 ,𝑛 𝐵,𝑖,𝑠 𝑖,𝐺 𝑖,𝑀 𝑖,𝑁 𝑅,𝑖 𝑖,𝑋 𝑖,𝑌 ↑ ,𝑖 · ,𝑏,𝑖 𝑇,𝑛 − ,𝑛 × ,𝑛 𝑖,𝑛 Allowed substitution hints: 𝐴(𝑖,𝑛,𝑠,𝑏) 𝐵(𝑏) 𝑃(𝑖,𝑛,𝑠,𝑏) + (𝑖,𝑛,𝑠,𝑏) 𝑅(𝑠,𝑏) 𝑇(𝑖,𝑠,𝑏) · (𝑛,𝑠) × (𝑖,𝑠,𝑏) ↑ (𝑛,𝑠,𝑏) 𝐺(𝑛,𝑠,𝑏) 𝑀(𝑠,𝑏) − (𝑖,𝑠,𝑏) 𝑁(𝑠,𝑏) 𝑋(𝑛,𝑠,𝑏) 𝑌(𝑠,𝑏) 0 (𝑖,𝑠,𝑏)

**Proof of Theorem chfacfscmulgsum**
Step	Hyp	Ref	Expression
1		eqid 2735	. . 3 ⊢ (Base‘𝑌) = (Base‘𝑌)
2		chfacfisf.0	. . 3 ⊢ 0 = (0_g‘𝑌)
3		chfacfscmulgsum.p	. . 3 ⊢ + = (+_g‘𝑌)
4		crngring 20215	. . . . . . . 8 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
5	4	anim2i 618	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
6	5	3adant3 1133	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
7		chfacfisf.p	. . . . . . 7 ⊢ 𝑃 = (Poly₁‘𝑅)
8		chfacfisf.y	. . . . . . 7 ⊢ 𝑌 = (𝑁 Mat 𝑃)
9	7, 8	pmatring 22645	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ Ring)
10	6, 9	syl 17	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Ring)
11		ringcmn 20252	. . . . 5 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ CMnd)
12	10, 11	syl 17	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ CMnd)
13	12	adantr 480	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑌 ∈ CMnd)
14		nn0ex 12432	. . . 4 ⊢ ℕ₀ ∈ V
15	14	a1i 11	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ℕ₀ ∈ V)
16		simpll 767	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ ℕ₀) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵))
17		simplr 769	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ ℕ₀) → (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))))
18		simpr 484	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ ℕ₀) → 𝑖 ∈ ℕ₀)
19	16, 17, 18	3jca 1129	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ ℕ₀) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝑖 ∈ ℕ₀))
20		chfacfisf.a	. . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅)
21		chfacfisf.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐴)
22		chfacfisf.r	. . . . 5 ⊢ × = (._r‘𝑌)
23		chfacfisf.s	. . . . 5 ⊢ − = (-_g‘𝑌)
24		chfacfisf.t	. . . . 5 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
25		chfacfisf.g	. . . . 5 ⊢ 𝐺 = (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))))
26		chfacfscmulcl.x	. . . . 5 ⊢ 𝑋 = (var₁‘𝑅)
27		chfacfscmulcl.m	. . . . 5 ⊢ · = ( ·_𝑠 ‘𝑌)
28		chfacfscmulcl.e	. . . . 5 ⊢ ↑ = (._g‘(mulGrp‘𝑃))
29	20, 21, 7, 8, 22, 23, 2, 24, 25, 26, 27, 28	chfacfscmulcl 22810	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝑖 ∈ ℕ₀) → ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) ∈ (Base‘𝑌))
30	19, 29	syl 17	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ ℕ₀) → ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) ∈ (Base‘𝑌))
31	20, 21, 7, 8, 22, 23, 2, 24, 25, 26, 27, 28	chfacfscmulfsupp 22812	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑖 ∈ ℕ₀ ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖))) finSupp 0 )
32		nn0disj 13587	. . . 4 ⊢ ((0...(𝑠 + 1)) ∩ (ℤ_≥‘((𝑠 + 1) + 1))) = ∅
33	32	a1i 11	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((0...(𝑠 + 1)) ∩ (ℤ_≥‘((𝑠 + 1) + 1))) = ∅)
34		nnnn0 12433	. . . . . 6 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ ℕ₀)
35		peano2nn0 12466	. . . . . 6 ⊢ (𝑠 ∈ ℕ₀ → (𝑠 + 1) ∈ ℕ₀)
36	34, 35	syl 17	. . . . 5 ⊢ (𝑠 ∈ ℕ → (𝑠 + 1) ∈ ℕ₀)
37		nn0split 13586	. . . . 5 ⊢ ((𝑠 + 1) ∈ ℕ₀ → ℕ₀ = ((0...(𝑠 + 1)) ∪ (ℤ_≥‘((𝑠 + 1) + 1))))
38	36, 37	syl 17	. . . 4 ⊢ (𝑠 ∈ ℕ → ℕ₀ = ((0...(𝑠 + 1)) ∪ (ℤ_≥‘((𝑠 + 1) + 1))))
39	38	ad2antrl 729	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ℕ₀ = ((0...(𝑠 + 1)) ∪ (ℤ_≥‘((𝑠 + 1) + 1))))
40	1, 2, 3, 13, 15, 30, 31, 33, 39	gsumsplit2 19893	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ ℕ₀ ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) = ((𝑌 Σ_g (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + (𝑌 Σ_g (𝑖 ∈ (ℤ_≥‘((𝑠 + 1) + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖))))))
41		simpll 767	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (ℤ_≥‘((𝑠 + 1) + 1))) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵))
42		simplr 769	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (ℤ_≥‘((𝑠 + 1) + 1))) → (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))))
43		nncn 12171	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ ℂ)
44		add1p1 12417	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ℂ → ((𝑠 + 1) + 1) = (𝑠 + 2))
45	43, 44	syl 17	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ ℕ → ((𝑠 + 1) + 1) = (𝑠 + 2))
46	45	ad2antrl 729	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑠 + 1) + 1) = (𝑠 + 2))
47	46	fveq2d 6833	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (ℤ_≥‘((𝑠 + 1) + 1)) = (ℤ_≥‘(𝑠 + 2)))
48	47	eleq2d 2821	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑖 ∈ (ℤ_≥‘((𝑠 + 1) + 1)) ↔ 𝑖 ∈ (ℤ_≥‘(𝑠 + 2))))
49	48	biimpa 476	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (ℤ_≥‘((𝑠 + 1) + 1))) → 𝑖 ∈ (ℤ_≥‘(𝑠 + 2)))
50	20, 21, 7, 8, 22, 23, 2, 24, 25, 26, 27, 28	chfacfscmul0 22811	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝑖 ∈ (ℤ_≥‘(𝑠 + 2))) → ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) = 0 )
51	41, 42, 49, 50	syl3anc 1374	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (ℤ_≥‘((𝑠 + 1) + 1))) → ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) = 0 )
52	51	mpteq2dva 5167	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑖 ∈ (ℤ_≥‘((𝑠 + 1) + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖))) = (𝑖 ∈ (ℤ_≥‘((𝑠 + 1) + 1)) ↦ 0 ))
53	52	oveq2d 7372	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (ℤ_≥‘((𝑠 + 1) + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) = (𝑌 Σ_g (𝑖 ∈ (ℤ_≥‘((𝑠 + 1) + 1)) ↦ 0 )))
54	4, 9	sylan2 594	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ Ring)
55		ringmnd 20213	. . . . . . . . . 10 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ Mnd)
56	54, 55	syl 17	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ Mnd)
57	56	3adant3 1133	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Mnd)
58		fvex 6842	. . . . . . . 8 ⊢ (ℤ_≥‘((𝑠 + 1) + 1)) ∈ V
59	57, 58	jctir 520	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑌 ∈ Mnd ∧ (ℤ_≥‘((𝑠 + 1) + 1)) ∈ V))
60	59	adantr 480	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 ∈ Mnd ∧ (ℤ_≥‘((𝑠 + 1) + 1)) ∈ V))
61	2	gsumz 18793	. . . . . 6 ⊢ ((𝑌 ∈ Mnd ∧ (ℤ_≥‘((𝑠 + 1) + 1)) ∈ V) → (𝑌 Σ_g (𝑖 ∈ (ℤ_≥‘((𝑠 + 1) + 1)) ↦ 0 )) = 0 )
62	60, 61	syl 17	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (ℤ_≥‘((𝑠 + 1) + 1)) ↦ 0 )) = 0 )
63	53, 62	eqtrd 2770	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (ℤ_≥‘((𝑠 + 1) + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) = 0 )
64	63	oveq2d 7372	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑌 Σ_g (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + (𝑌 Σ_g (𝑖 ∈ (ℤ_≥‘((𝑠 + 1) + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖))))) = ((𝑌 Σ_g (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + 0 ))
65		fzfid 13924	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (0...(𝑠 + 1)) ∈ Fin)
66		elfznn0 13563	. . . . . . . 8 ⊢ (𝑖 ∈ (0...(𝑠 + 1)) → 𝑖 ∈ ℕ₀)
67	66, 19	sylan2 594	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...(𝑠 + 1))) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝑖 ∈ ℕ₀))
68	67, 29	syl 17	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...(𝑠 + 1))) → ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) ∈ (Base‘𝑌))
69	68	ralrimiva 3127	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ∀𝑖 ∈ (0...(𝑠 + 1))((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) ∈ (Base‘𝑌))
70	1, 13, 65, 69	gsummptcl 19931	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) ∈ (Base‘𝑌))
71	1, 3, 2	mndrid 18712	. . . 4 ⊢ ((𝑌 ∈ Mnd ∧ (𝑌 Σ_g (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) ∈ (Base‘𝑌)) → ((𝑌 Σ_g (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + 0 ) = (𝑌 Σ_g (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))))
72	57, 70, 71	syl2an2r 686	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑌 Σ_g (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + 0 ) = (𝑌 Σ_g (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))))
73	64, 72	eqtrd 2770	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑌 Σ_g (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + (𝑌 Σ_g (𝑖 ∈ (ℤ_≥‘((𝑠 + 1) + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖))))) = (𝑌 Σ_g (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))))
74	34	ad2antrl 729	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑠 ∈ ℕ₀)
75	1, 3, 13, 74, 68	gsummptfzsplit 19896	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) = ((𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + (𝑌 Σ_g (𝑖 ∈ {(𝑠 + 1)} ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖))))))
76		elfznn0 13563	. . . . . . 7 ⊢ (𝑖 ∈ (0...𝑠) → 𝑖 ∈ ℕ₀)
77	76, 30	sylan2 594	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) ∈ (Base‘𝑌))
78	1, 3, 13, 74, 77	gsummptfzsplitl 19897	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + (𝑌 Σ_g (𝑖 ∈ {0} ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖))))))
79	57	adantr 480	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑌 ∈ Mnd)
80		0nn0 12441	. . . . . . . 8 ⊢ 0 ∈ ℕ₀
81	80	a1i 11	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 0 ∈ ℕ₀)
82	20, 21, 7, 8, 22, 23, 2, 24, 25, 26, 27, 28	chfacfscmulcl 22810	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 0 ∈ ℕ₀) → ((0 ↑ 𝑋) · (𝐺‘0)) ∈ (Base‘𝑌))
83	81, 82	mpd3an3 1465	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((0 ↑ 𝑋) · (𝐺‘0)) ∈ (Base‘𝑌))
84		oveq1 7363	. . . . . . . . 9 ⊢ (𝑖 = 0 → (𝑖 ↑ 𝑋) = (0 ↑ 𝑋))
85		fveq2 6829	. . . . . . . . 9 ⊢ (𝑖 = 0 → (𝐺‘𝑖) = (𝐺‘0))
86	84, 85	oveq12d 7374	. . . . . . . 8 ⊢ (𝑖 = 0 → ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) = ((0 ↑ 𝑋) · (𝐺‘0)))
87	1, 86	gsumsn 19918	. . . . . . 7 ⊢ ((𝑌 ∈ Mnd ∧ 0 ∈ ℕ₀ ∧ ((0 ↑ 𝑋) · (𝐺‘0)) ∈ (Base‘𝑌)) → (𝑌 Σ_g (𝑖 ∈ {0} ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) = ((0 ↑ 𝑋) · (𝐺‘0)))
88	79, 81, 83, 87	syl3anc 1374	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ {0} ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) = ((0 ↑ 𝑋) · (𝐺‘0)))
89	88	oveq2d 7372	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + (𝑌 Σ_g (𝑖 ∈ {0} ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖))))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + ((0 ↑ 𝑋) · (𝐺‘0))))
90	78, 89	eqtrd 2770	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + ((0 ↑ 𝑋) · (𝐺‘0))))
91		ovexd 7391	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑠 + 1) ∈ V)
92		1nn0 12442	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
93	92	a1i 11	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 1 ∈ ℕ₀)
94	74, 93	nn0addcld 12491	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑠 + 1) ∈ ℕ₀)
95	20, 21, 7, 8, 22, 23, 2, 24, 25, 26, 27, 28	chfacfscmulcl 22810	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ (𝑠 + 1) ∈ ℕ₀) → (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1))) ∈ (Base‘𝑌))
96	94, 95	mpd3an3 1465	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1))) ∈ (Base‘𝑌))
97		oveq1 7363	. . . . . . 7 ⊢ (𝑖 = (𝑠 + 1) → (𝑖 ↑ 𝑋) = ((𝑠 + 1) ↑ 𝑋))
98		fveq2 6829	. . . . . . 7 ⊢ (𝑖 = (𝑠 + 1) → (𝐺‘𝑖) = (𝐺‘(𝑠 + 1)))
99	97, 98	oveq12d 7374	. . . . . 6 ⊢ (𝑖 = (𝑠 + 1) → ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) = (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1))))
100	1, 99	gsumsn 19918	. . . . 5 ⊢ ((𝑌 ∈ Mnd ∧ (𝑠 + 1) ∈ V ∧ (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1))) ∈ (Base‘𝑌)) → (𝑌 Σ_g (𝑖 ∈ {(𝑠 + 1)} ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) = (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1))))
101	79, 91, 96, 100	syl3anc 1374	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ {(𝑠 + 1)} ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) = (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1))))
102	90, 101	oveq12d 7374	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + (𝑌 Σ_g (𝑖 ∈ {(𝑠 + 1)} ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖))))) = (((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + ((0 ↑ 𝑋) · (𝐺‘0))) + (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1)))))
103		fzfid 13924	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (1...𝑠) ∈ Fin)
104		simpll 767	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵))
105		simplr 769	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))))
106		elfznn 13496	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1...𝑠) → 𝑖 ∈ ℕ)
107	106	nnnn0d 12487	. . . . . . . . 9 ⊢ (𝑖 ∈ (1...𝑠) → 𝑖 ∈ ℕ₀)
108	107	adantl 481	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → 𝑖 ∈ ℕ₀)
109	104, 105, 108, 29	syl3anc 1374	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) ∈ (Base‘𝑌))
110	109	ralrimiva 3127	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ∀𝑖 ∈ (1...𝑠)((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) ∈ (Base‘𝑌))
111	1, 13, 103, 110	gsummptcl 19931	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) ∈ (Base‘𝑌))
112	1, 3	mndass 18700	. . . . 5 ⊢ ((𝑌 ∈ Mnd ∧ ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) ∈ (Base‘𝑌) ∧ ((0 ↑ 𝑋) · (𝐺‘0)) ∈ (Base‘𝑌) ∧ (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1))) ∈ (Base‘𝑌))) → (((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + ((0 ↑ 𝑋) · (𝐺‘0))) + (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1)))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + (((0 ↑ 𝑋) · (𝐺‘0)) + (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1))))))
113	79, 111, 83, 96, 112	syl13anc 1375	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + ((0 ↑ 𝑋) · (𝐺‘0))) + (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1)))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + (((0 ↑ 𝑋) · (𝐺‘0)) + (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1))))))
114	106	nnne0d 12216	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (1...𝑠) → 𝑖 ≠ 0)
115	114	ad2antlr 728	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑖 ≠ 0)
116		neeq1 2992	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑖 → (𝑛 ≠ 0 ↔ 𝑖 ≠ 0))
117	116	adantl 481	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → (𝑛 ≠ 0 ↔ 𝑖 ≠ 0))
118	115, 117	mpbird 257	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑛 ≠ 0)
119		eqneqall 2941	. . . . . . . . . . . 12 ⊢ (𝑛 = 0 → (𝑛 ≠ 0 → 0 = (𝑇‘(𝑏‘(𝑖 − 1)))))
120	118, 119	mpan9 506	. . . . . . . . . . 11 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → 0 = (𝑇‘(𝑏‘(𝑖 − 1))))
121		simplr 769	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → 𝑛 = 𝑖)
122		eqeq1 2739	. . . . . . . . . . . . . . . . 17 ⊢ (0 = 𝑛 → (0 = 𝑖 ↔ 𝑛 = 𝑖))
123	122	eqcoms 2743	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 0 → (0 = 𝑖 ↔ 𝑛 = 𝑖))
124	123	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → (0 = 𝑖 ↔ 𝑛 = 𝑖))
125	121, 124	mpbird 257	. . . . . . . . . . . . . 14 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → 0 = 𝑖)
126	125	fveq2d 6833	. . . . . . . . . . . . 13 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → (𝑏‘0) = (𝑏‘𝑖))
127	126	fveq2d 6833	. . . . . . . . . . . 12 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → (𝑇‘(𝑏‘0)) = (𝑇‘(𝑏‘𝑖)))
128	127	oveq2d 7372	. . . . . . . . . . 11 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))) = ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))
129	120, 128	oveq12d 7374	. . . . . . . . . 10 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) = ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))
130		elfz2 13457	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ (1...𝑠) ↔ ((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖 ∧ 𝑖 ≤ 𝑠)))
131		zleltp1 12567	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑖 ∈ ℤ ∧ 𝑠 ∈ ℤ) → (𝑖 ≤ 𝑠 ↔ 𝑖 < (𝑠 + 1)))
132	131	ancoms 458	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑖 ≤ 𝑠 ↔ 𝑖 < (𝑠 + 1)))
133	132	3adant1 1131	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑖 ≤ 𝑠 ↔ 𝑖 < (𝑠 + 1)))
134	133	biimpcd 249	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 ≤ 𝑠 → ((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → 𝑖 < (𝑠 + 1)))
135	134	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1 ≤ 𝑖 ∧ 𝑖 ≤ 𝑠) → ((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → 𝑖 < (𝑠 + 1)))
136	135	impcom 407	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖 ∧ 𝑖 ≤ 𝑠)) → 𝑖 < (𝑠 + 1))
137	136	orcd 874	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖 ∧ 𝑖 ≤ 𝑠)) → (𝑖 < (𝑠 + 1) ∨ (𝑠 + 1) < 𝑖))
138		zre 12517	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑠 ∈ ℤ → 𝑠 ∈ ℝ)
139		1red 11134	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑠 ∈ ℤ → 1 ∈ ℝ)
140	138, 139	readdcld 11163	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑠 ∈ ℤ → (𝑠 + 1) ∈ ℝ)
141		zre 12517	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 ∈ ℤ → 𝑖 ∈ ℝ)
142	140, 141	anim12ci 615	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑖 ∈ ℝ ∧ (𝑠 + 1) ∈ ℝ))
143	142	3adant1 1131	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑖 ∈ ℝ ∧ (𝑠 + 1) ∈ ℝ))
144		lttri2 11217	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ ℝ ∧ (𝑠 + 1) ∈ ℝ) → (𝑖 ≠ (𝑠 + 1) ↔ (𝑖 < (𝑠 + 1) ∨ (𝑠 + 1) < 𝑖)))
145	143, 144	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑖 ≠ (𝑠 + 1) ↔ (𝑖 < (𝑠 + 1) ∨ (𝑠 + 1) < 𝑖)))
146	145	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖 ∧ 𝑖 ≤ 𝑠)) → (𝑖 ≠ (𝑠 + 1) ↔ (𝑖 < (𝑠 + 1) ∨ (𝑠 + 1) < 𝑖)))
147	137, 146	mpbird 257	. . . . . . . . . . . . . . . . . 18 ⊢ (((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖 ∧ 𝑖 ≤ 𝑠)) → 𝑖 ≠ (𝑠 + 1))
148	130, 147	sylbi 217	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (1...𝑠) → 𝑖 ≠ (𝑠 + 1))
149	148	ad2antlr 728	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑖 ≠ (𝑠 + 1))
150		neeq1 2992	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑖 → (𝑛 ≠ (𝑠 + 1) ↔ 𝑖 ≠ (𝑠 + 1)))
151	150	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → (𝑛 ≠ (𝑠 + 1) ↔ 𝑖 ≠ (𝑠 + 1)))
152	149, 151	mpbird 257	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑛 ≠ (𝑠 + 1))
153	152	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) → 𝑛 ≠ (𝑠 + 1))
154	153	neneqd 2935	. . . . . . . . . . . . 13 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) → ¬ 𝑛 = (𝑠 + 1))
155	154	pm2.21d 121	. . . . . . . . . . . 12 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) → (𝑛 = (𝑠 + 1) → (𝑇‘(𝑏‘𝑠)) = ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))
156	155	imp 406	. . . . . . . . . . 11 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ 𝑛 = (𝑠 + 1)) → (𝑇‘(𝑏‘𝑠)) = ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))
157	106	nnred 12178	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ (1...𝑠) → 𝑖 ∈ ℝ)
158		eleq1w 2818	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑖 → (𝑛 ∈ ℝ ↔ 𝑖 ∈ ℝ))
159	157, 158	syl5ibrcom 247	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ (1...𝑠) → (𝑛 = 𝑖 → 𝑛 ∈ ℝ))
160	159	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑛 = 𝑖 → 𝑛 ∈ ℝ))
161	160	imp 406	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑛 ∈ ℝ)
162	74	nn0red 12488	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑠 ∈ ℝ)
163	162	ad2antrr 727	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑠 ∈ ℝ)
164		1red 11134	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 1 ∈ ℝ)
165	163, 164	readdcld 11163	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → (𝑠 + 1) ∈ ℝ)
166	130, 136	sylbi 217	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ (1...𝑠) → 𝑖 < (𝑠 + 1))
167	166	ad2antlr 728	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑖 < (𝑠 + 1))
168		breq1 5077	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑖 → (𝑛 < (𝑠 + 1) ↔ 𝑖 < (𝑠 + 1)))
169	168	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → (𝑛 < (𝑠 + 1) ↔ 𝑖 < (𝑠 + 1)))
170	167, 169	mpbird 257	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑛 < (𝑠 + 1))
171	161, 165, 170	ltnsymd 11284	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → ¬ (𝑠 + 1) < 𝑛)
172	171	pm2.21d 121	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → ((𝑠 + 1) < 𝑛 → 0 = ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))
173	172	ad2antrr 727	. . . . . . . . . . . . 13 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) → ((𝑠 + 1) < 𝑛 → 0 = ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))
174	173	imp 406	. . . . . . . . . . . 12 ⊢ ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ (𝑠 + 1) < 𝑛) → 0 = ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))
175		simp-4r 784	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → 𝑛 = 𝑖)
176	175	fvoveq1d 7378	. . . . . . . . . . . . . 14 ⊢ ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → (𝑏‘(𝑛 − 1)) = (𝑏‘(𝑖 − 1)))
177	176	fveq2d 6833	. . . . . . . . . . . . 13 ⊢ ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → (𝑇‘(𝑏‘(𝑛 − 1))) = (𝑇‘(𝑏‘(𝑖 − 1))))
178	175	fveq2d 6833	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → (𝑏‘𝑛) = (𝑏‘𝑖))
179	178	fveq2d 6833	. . . . . . . . . . . . . 14 ⊢ ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → (𝑇‘(𝑏‘𝑛)) = (𝑇‘(𝑏‘𝑖)))
180	179	oveq2d 7372	. . . . . . . . . . . . 13 ⊢ ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))) = ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))
181	177, 180	oveq12d 7374	. . . . . . . . . . . 12 ⊢ ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))) = ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))
182	174, 181	ifeqda 4493	. . . . . . . . . . 11 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) → if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))) = ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))
183	156, 182	ifeqda 4493	. . . . . . . . . 10 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) → if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))) = ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))
184	129, 183	ifeqda 4493	. . . . . . . . 9 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))) = ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))
185		ovexd 7391	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))) ∈ V)
186	25, 184, 108, 185	fvmptd2 6945	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝐺‘𝑖) = ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))
187	186	oveq2d 7372	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) = ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))
188	187	mpteq2dva 5167	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖))) = (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))))
189	188	oveq2d 7372	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) = (𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))))
190		nn0p1gt0 12455	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ ℕ₀ → 0 < (𝑠 + 1))
191		0red 11136	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ ℕ₀ → 0 ∈ ℝ)
192		ltne 11232	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℝ ∧ 0 < (𝑠 + 1)) → (𝑠 + 1) ≠ 0)
193	191, 192	sylan 581	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 ∈ ℕ₀ ∧ 0 < (𝑠 + 1)) → (𝑠 + 1) ≠ 0)
194		neeq1 2992	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (𝑠 + 1) → (𝑛 ≠ 0 ↔ (𝑠 + 1) ≠ 0))
195	193, 194	syl5ibrcom 247	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ ℕ₀ ∧ 0 < (𝑠 + 1)) → (𝑛 = (𝑠 + 1) → 𝑛 ≠ 0))
196	34, 190, 195	syl2anc2 586	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ℕ → (𝑛 = (𝑠 + 1) → 𝑛 ≠ 0))
197	196	ad2antrl 729	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑛 = (𝑠 + 1) → 𝑛 ≠ 0))
198	197	imp 406	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑛 = (𝑠 + 1)) → 𝑛 ≠ 0)
199		eqneqall 2941	. . . . . . . . . . 11 ⊢ (𝑛 = 0 → (𝑛 ≠ 0 → ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) = (𝑇‘(𝑏‘𝑠))))
200	198, 199	mpan9 506	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑛 = (𝑠 + 1)) ∧ 𝑛 = 0) → ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) = (𝑇‘(𝑏‘𝑠)))
201		iftrue 4462	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑠 + 1) → if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))) = (𝑇‘(𝑏‘𝑠)))
202	201	ad2antlr 728	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑛 = (𝑠 + 1)) ∧ ¬ 𝑛 = 0) → if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))) = (𝑇‘(𝑏‘𝑠)))
203	200, 202	ifeqda 4493	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑛 = (𝑠 + 1)) → if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))) = (𝑇‘(𝑏‘𝑠)))
204	74, 35	syl 17	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑠 + 1) ∈ ℕ₀)
205		fvexd 6844	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑇‘(𝑏‘𝑠)) ∈ V)
206	25, 203, 204, 205	fvmptd2 6945	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝐺‘(𝑠 + 1)) = (𝑇‘(𝑏‘𝑠)))
207	206	oveq2d 7372	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1))) = (((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))))
208	4	3ad2ant2 1135	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring)
209		eqid 2735	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑃) = (Base‘𝑃)
210	26, 7, 209	vr1cl 22169	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ (Base‘𝑃))
211	208, 210	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑋 ∈ (Base‘𝑃))
212		eqid 2735	. . . . . . . . . . . . . 14 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
213	212, 209	mgpbas 20115	. . . . . . . . . . . . 13 ⊢ (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
214		eqid 2735	. . . . . . . . . . . . . 14 ⊢ (1_r‘𝑃) = (1_r‘𝑃)
215	212, 214	ringidval 20153	. . . . . . . . . . . . 13 ⊢ (1_r‘𝑃) = (0_g‘(mulGrp‘𝑃))
216	213, 215, 28	mulg0 19039	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ (Base‘𝑃) → (0 ↑ 𝑋) = (1_r‘𝑃))
217	211, 216	syl 17	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (0 ↑ 𝑋) = (1_r‘𝑃))
218	7	ply1crng 22150	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ CRing)
219	218	anim2i 618	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑃 ∈ CRing))
220	219	3adant3 1133	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑃 ∈ CRing))
221	8	matsca2 22373	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ CRing) → 𝑃 = (Scalar‘𝑌))
222	220, 221	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑃 = (Scalar‘𝑌))
223	222	fveq2d 6833	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (1_r‘𝑃) = (1_r‘(Scalar‘𝑌)))
224	217, 223	eqtrd 2770	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (0 ↑ 𝑋) = (1_r‘(Scalar‘𝑌)))
225	224	adantr 480	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (0 ↑ 𝑋) = (1_r‘(Scalar‘𝑌)))
226	225	oveq1d 7371	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((0 ↑ 𝑋) · (𝐺‘0)) = ((1_r‘(Scalar‘𝑌)) · (𝐺‘0)))
227	7, 8	pmatlmod 22646	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ LMod)
228	4, 227	sylan2 594	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ LMod)
229	228	3adant3 1133	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ LMod)
230	20, 21, 7, 8, 22, 23, 2, 24, 25	chfacfisf 22807	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝐺:ℕ₀⟶(Base‘𝑌))
231	4, 230	syl3anl2 1416	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝐺:ℕ₀⟶(Base‘𝑌))
232	231, 81	ffvelcdmd 7026	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝐺‘0) ∈ (Base‘𝑌))
233		eqid 2735	. . . . . . . . . 10 ⊢ (Scalar‘𝑌) = (Scalar‘𝑌)
234		eqid 2735	. . . . . . . . . 10 ⊢ (1_r‘(Scalar‘𝑌)) = (1_r‘(Scalar‘𝑌))
235	1, 233, 27, 234	lmodvs1 20874	. . . . . . . . 9 ⊢ ((𝑌 ∈ LMod ∧ (𝐺‘0) ∈ (Base‘𝑌)) → ((1_r‘(Scalar‘𝑌)) · (𝐺‘0)) = (𝐺‘0))
236	229, 232, 235	syl2an2r 686	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((1_r‘(Scalar‘𝑌)) · (𝐺‘0)) = (𝐺‘0))
237		iftrue 4462	. . . . . . . . 9 ⊢ (𝑛 = 0 → if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))) = ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))))
238		ovexd 7391	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) ∈ V)
239	25, 237, 81, 238	fvmptd3 6960	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝐺‘0) = ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))))
240	226, 236, 239	3eqtrd 2774	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((0 ↑ 𝑋) · (𝐺‘0)) = ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))))
241	207, 240	oveq12d 7374	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1))) + ((0 ↑ 𝑋) · (𝐺‘0))) = ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) + ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))
242	1, 3	cmncom 19762	. . . . . . 7 ⊢ ((𝑌 ∈ CMnd ∧ ((0 ↑ 𝑋) · (𝐺‘0)) ∈ (Base‘𝑌) ∧ (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1))) ∈ (Base‘𝑌)) → (((0 ↑ 𝑋) · (𝐺‘0)) + (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1)))) = ((((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1))) + ((0 ↑ 𝑋) · (𝐺‘0))))
243	13, 83, 96, 242	syl3anc 1374	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (((0 ↑ 𝑋) · (𝐺‘0)) + (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1)))) = ((((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1))) + ((0 ↑ 𝑋) · (𝐺‘0))))
244		ringgrp 20208	. . . . . . . . 9 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ Grp)
245	10, 244	syl 17	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Grp)
246	245	adantr 480	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑌 ∈ Grp)
247	207, 96	eqeltrrd 2836	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) ∈ (Base‘𝑌))
248	10	adantr 480	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑌 ∈ Ring)
249	24, 20, 21, 7, 8	mat2pmatbas 22679	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝑌))
250	4, 249	syl3an2 1165	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝑌))
251	250	adantr 480	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑇‘𝑀) ∈ (Base‘𝑌))
252		simpl1 1193	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑁 ∈ Fin)
253	208	adantr 480	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑅 ∈ Ring)
254		elmapi 8785	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ (𝐵 ↑_m (0...𝑠)) → 𝑏:(0...𝑠)⟶𝐵)
255	254	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → 𝑏:(0...𝑠)⟶𝐵)
256	255	adantl 481	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑏:(0...𝑠)⟶𝐵)
257		0elfz 13567	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ ℕ₀ → 0 ∈ (0...𝑠))
258	34, 257	syl 17	. . . . . . . . . . 11 ⊢ (𝑠 ∈ ℕ → 0 ∈ (0...𝑠))
259	258	ad2antrl 729	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 0 ∈ (0...𝑠))
260	256, 259	ffvelcdmd 7026	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑏‘0) ∈ 𝐵)
261	24, 20, 21, 7, 8	mat2pmatbas 22679	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘0) ∈ 𝐵) → (𝑇‘(𝑏‘0)) ∈ (Base‘𝑌))
262	252, 253, 260, 261	syl3anc 1374	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑇‘(𝑏‘0)) ∈ (Base‘𝑌))
263	1, 22	ringcl 20220	. . . . . . . 8 ⊢ ((𝑌 ∈ Ring ∧ (𝑇‘𝑀) ∈ (Base‘𝑌) ∧ (𝑇‘(𝑏‘0)) ∈ (Base‘𝑌)) → ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))) ∈ (Base‘𝑌))
264	248, 251, 262, 263	syl3anc 1374	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))) ∈ (Base‘𝑌))
265	1, 2, 23, 3	grpsubadd0sub 18992	. . . . . . 7 ⊢ ((𝑌 ∈ Grp ∧ (((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) ∈ (Base‘𝑌) ∧ ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))) ∈ (Base‘𝑌)) → ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) = ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) + ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))
266	246, 247, 264, 265	syl3anc 1374	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) = ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) + ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))
267	241, 243, 266	3eqtr4d 2780	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (((0 ↑ 𝑋) · (𝐺‘0)) + (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1)))) = ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))))
268	189, 267	oveq12d 7374	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + (((0 ↑ 𝑋) · (𝐺‘0)) + (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1))))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))
269	113, 268	eqtrd 2770	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + ((0 ↑ 𝑋) · (𝐺‘0))) + (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1)))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))
270	75, 102, 269	3eqtrd 2774	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))
271	40, 73, 270	3eqtrd 2774	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ ℕ₀ ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2930 Vcvv 3427 ∪ cun 3883 ∩ cin 3884 ∅c0 4263 ifcif 4456 {csn 4557 class class class wbr 5074 ↦ cmpt 5155 ⟶wf 6483 ‘cfv 6487 (class class class)co 7356 ↑_m cmap 8762 Fincfn 8882 ℂcc 11025 ℝcr 11026 0cc0 11027 1c1 11028 + caddc 11030 < clt 11168 ≤ cle 11169 − cmin 11366 ℕcn 12163 2c2 12225 ℕ₀cn0 12426 ℤcz 12513 ℤ_≥cuz 12777 ...cfz 13450 Basecbs 17168 +_gcplusg 17209 ._rcmulr 17210 Scalarcsca 17212 ·_𝑠 cvsca 17213 0_gc0g 17391 Σ_g cgsu 17392 Mndcmnd 18691 Grpcgrp 18898 -_gcsg 18900 ._gcmg 19032 CMndccmn 19744 mulGrpcmgp 20110 1_rcur 20151 Ringcrg 20203 CRingccrg 20204 LModclmod 20844 var₁cv1 22128 Poly₁cpl1 22129 Mat cmat 22360 matToPolyMat cmat2pmat 22657

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2184 ax-ext 2707 ax-rep 5201 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7678 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3060 df-rmo 3340 df-reu 3341 df-rab 3388 df-v 3429 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-ot 4566 df-uni 4841 df-int 4880 df-iun 4925 df-iin 4926 df-br 5075 df-opab 5137 df-mpt 5156 df-tr 5182 df-id 5515 df-eprel 5520 df-po 5528 df-so 5529 df-fr 5573 df-se 5574 df-we 5575 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6254 df-ord 6315 df-on 6316 df-lim 6317 df-suc 6318 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-isom 6496 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-ofr 7621 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8100 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-2o 8395 df-er 8632 df-map 8764 df-pm 8765 df-ixp 8835 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-fsupp 9264 df-sup 9344 df-oi 9414 df-card 9852 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-nn 12164 df-2 12233 df-3 12234 df-4 12235 df-5 12236 df-6 12237 df-7 12238 df-8 12239 df-9 12240 df-n0 12427 df-z 12514 df-dec 12634 df-uz 12778 df-rp 12932 df-fz 13451 df-fzo 13598 df-seq 13953 df-hash 14282 df-struct 17106 df-sets 17123 df-slot 17141 df-ndx 17153 df-base 17169 df-ress 17190 df-plusg 17222 df-mulr 17223 df-sca 17225 df-vsca 17226 df-ip 17227 df-tset 17228 df-ple 17229 df-ds 17231 df-hom 17233 df-cco 17234 df-0g 17393 df-gsum 17394 df-prds 17399 df-pws 17401 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-mhm 18740 df-submnd 18741 df-grp 18901 df-minusg 18902 df-sbg 18903 df-mulg 19033 df-subg 19088 df-ghm 19177 df-cntz 19281 df-cmn 19746 df-abl 19747 df-mgp 20111 df-rng 20123 df-ur 20152 df-ring 20205 df-cring 20206 df-subrng 20512 df-subrg 20536 df-lmod 20846 df-lss 20916 df-sra 21157 df-rgmod 21158 df-dsmm 21701 df-frlm 21716 df-ascl 21824 df-psr 21878 df-mvr 21879 df-mpl 21880 df-opsr 21882 df-psr1 22132 df-vr1 22133 df-ply1 22134 df-mamu 22344 df-mat 22361 df-mat2pmat 22660

This theorem is referenced by: cpmadugsum 22831

W3C validator